Suppose that a certain linear system transforms an incoming signal $f$ into an outgoing signal $y$ that is a solution of $$y''(t)+ay'(t)+by(t) = f(t)$$ where $a$ and $b$ are constants. Show that if the roots of the characteristic equation $r^2+ar+b =0$ both have their real parts $<0$, then the system is causal.
I know that the system is causal if the value $y(t)$ at any time $t$ depends only on the values of $f(u)$ for $u\leq t$. This reminds me of convolution, which is a causal operator. But finding a function $g(t)$ such that $y(t) = [g*f](t)$. Any ideas?